206 GROWTH PRINCIPLES AND THEORY 2 



also in curves of length growth; only, e.g. in fish, it is so close to the origin of the 

 coordinates that the remaining curve can be approximated by Bertalanffy's 

 equation. The answer to this objection is that obviously the theory is applicable 

 only insofar as its assumptions, i.e. an increase in size without far reaching changes 

 of anatomic and physiological organization, hold true as a gross approximation. 

 This is not the case in embryonic and larval development. 



ib) The growth equations of Bertalanffy (and similarly almost all other 

 equations proposed) state that the final v^lue be reached asymptotically, i.e. 

 "in infinite time". This is a simplification which does not strictly correspond to 

 the facts, because many animals terminate growth within finite time. An illus- 

 tration is the comparison of the mouse which terminates growth soon, and the 

 rat, which grows during the whole life span, at least insofar as growth is defined as 

 increase in weight. In other terms, the equations assumed disregard a factor of 

 ageing which, at least in certain cases, is not negligible. The reason is essentially 

 a mathematical one: for functions asymptotically tending to a limiting value are 

 simpler and easier to handle than functions reaching such value in finite time. 



There is a contrast of animals with indeterminate {e.g. fish) and terminate growth 

 (e.g. mammals and also, based upon a different mechanism, insects), i.e. such 

 which are growing during their whole life span and such where growth stops at 

 a certain time. Two alternative hypotheses are possible, namely, that there is an 

 essential difference between these two types, or that the course of growth is 

 essentially the same and secondary factors such as puberty and senescence lead 

 to termination of growth. At present there seems to be no physiological basis for 

 deciding between these alternatives, and for reasons of mathematical expedience 

 the second was adopted. In this we are in agreeinent with D'Ancona (1952), 

 according to whom the distinction between indeterminate and terminate growth, 

 as in fish and mammals 



"is more apparent than real . . . This difference essentially depends on a different relation 

 between life span and period of growth proper, upon which sexual maturation certainly 

 has a preeminent influence. Thus it can be maintained at present that for fish and other 

 species with indeterminate growth, little is known about the efTective potential duration 

 of life, and true phenomena of senescence are rarely encountered." 



It is well possible that the "constants" of the growth equations which express 

 the interplay of growth-promoting and growth-inhibiting factors actually are not 

 constant but change during the life cycle. Physiologically, this would mean that 

 anabolic and catabolic factors change, and processes of ageing should be taken 

 into account. The mathematical expression of this would be that •/] and x are 

 functions of time and/or additional terms should be introduced into the growth 

 equations. However, inasmuch as the simple growth model covers a wide range 

 of data and accessory hypotheses would be gratuitous, it is of little value to 

 complicate the mathematical structure so long as this is not based upon physiolog- 

 ical and experimental evidence. It would not be difficult to account for senescence 

 and termination of growth by introducing terms in time into the growth equation 

 (5.18); but the resulting complication of formulas and introduction of new 

 constants would make the improvement so obtained somewhat specious, and 



